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## Asymptotic properties of third order functional dynamic equations on time scales

### Tom 100 / 2011

Annales Polonici Mathematici 100 (2011), 203-222 MSC: 34K11, 39A10, 39A99. DOI: 10.4064/ap100-3-1

#### Streszczenie

The purpose of this paper is to study the asymptotic properties of nonoscillatory solutions of the third order nonlinear functional dynamic equation $$[ p(t)[ (r(t)x^{\Delta }(t))^{\Delta }] ^{\gamma }] ^{\Delta }+q(t)f(x(\tau (t)))=0,\quad\ t\geq t_{0},$$ on a time scale $\mathbb{T}$, where $\gamma >0$ is a quotient of odd positive integers, and $p$, $q$, $r$ and $\tau$ are positive right-dense continuous functions defined on $\mathbb{T}$. We classify the nonoscillatory solutions into certain classes $C_{i}$, $i=0,1,2,3$, according to the sign of the $\Delta$-quasi-derivatives and obtain sufficient conditions in order that $C_{i}=\emptyset .$ Also, we establish some sufficient conditions which ensure the property $A$ of the solutions. Our results are new for third order dynamic equations and involve and improve some results previously obtained for differential and difference equations. Some examples are worked out to demonstrate the main results.

#### Autorzy

• I. KubiaczykFaculty of Mathematics and Computer Science
Umultowska 87
61-614 Poznań, Poland
e-mail
• S. H. SakerDepartment of Mathematics Skills, PY
King Saud University
and
Department of Mathematics
Faculty of Science
Mansoura University
Mansoura, 35516, Egypt
e-mail
e-mail

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